Time-Changed Compound Poisson-Gamma Subordinators

• The paper demonstrates how time-changing a Poisson process with a Gamma subordinator yields tractable finite-activity jump processes.
• It formulates explicit Laplace exponents and Lévy measures that capture clustered dynamics and provide analytical insights into bursty events.
• The work connects these subordinators with anomalous diffusion and non-local governing equations, offering practical tools for stochastic modeling.

A time-changed compound Poisson–Gamma subordinator is a specific class of stochastic process constructed by employing the Gamma subordinator with Poissonian randomization, and then utilizing this process to randomize the clock (i.e., the time argument) of another process, most often a Lévy or Poisson process. This framework unifies finite-activity jump processes with subordinated dynamics and is essential in modeling bursty or clustered phenomena in a range of probabilistic systems, with deep connections to anomalous diffusion, aggregated stochastic modeling, and non-local governing equations (Buchak et al., 2017, Buchak et al., 2018, Beghin et al., 2021).

1. Definition and Construction

Let $N_\lambda(t)$ be a Poisson process with rate $\lambda > 0$. For a Gamma subordinator $G_{\alpha,\beta}(s)$ — with shape parameter $\alpha > 0$ and rate $\beta > 0$ — the compound Poisson–Gamma subordinator is defined as

$$S(t) = \sum_{n=1}^{N_\lambda(t)} G_n \equiv G(N_\lambda(t)), \quad S(0) = 0,$$

where $\{G_n\}$ are i.i.d. Gamma($\alpha,\beta$) random variables [(Buchak et al., 2017), Sec. 3]. The resulting process $S(t)$ is a pure-jump subordinator with finite activity.

When this subordinator is used to direct (“time-change”) another independent stochastic process — often a Poisson process $N_{\lambda_1}$ — one obtains processes such as $X(t) = N_{\lambda_1}(S(t))$. More generally, one can replace the outer process with any Lévy process (Buchak et al., 2017, Buchak et al., 2018).

2. Analytical Properties

Laplace Exponent and Lévy Measure

The Lévy–Khintchine exponent of the compound Poisson–Gamma subordinator is

$$\psi(\theta) = -\log \mathbb{E} [e^{-\theta S(1)}] = \lambda \beta^\alpha\left[\beta^{-\alpha} - (\beta+\theta)^{-\alpha}\right].$$

Its Lévy measure is

$$\nu(du) = \lambda \frac{\beta^\alpha}{\Gamma(\alpha)} u^{\alpha-1} e^{-\beta u}\, du.$$

For $\alpha=1$, $S(t)$ becomes a compound Poisson process with exponential jumps [(Buchak et al., 2017), (3.1)–(3.2)].

Marginal Distribution

For $t > 0$,

$$P\{S(t) = 0\} = e^{-\lambda t}, \quad P\{S(t) \in ds\} = e^{-\lambda t} \delta_0(ds) + e^{-\lambda t - \beta s} s^{-1} \Phi(\alpha, 0; \lambda t (\beta s)^\alpha)\, ds$$

where $\Phi$ is the Wright function [(Buchak et al., 2017), (3.2)]. Alternatively,

$$f_{S(t)}(s) = e^{-\lambda t - \beta s} \sum_{n=1}^\infty \frac{(\lambda t \beta^\alpha)^n}{n! \Gamma(\alpha n)} s^{\alpha n - 1}.$$

3. Time-Changed Poisson Processes

Distributional Law

Letting $X(t) = N_{\lambda_1}(S(t))$, the marginal law is given by

$$p_k(t) = P\{X(t) = k\} = e^{-\lambda t} \frac{k! \lambda_1^k}{(\lambda_1+\beta)^k} \sum_{n=1}^\infty \frac{(\lambda t \beta^\alpha)^n\,\Gamma(\alpha n + k)}{n! \Gamma(\alpha n) (\lambda_1+\beta)^{\alpha n}}, \quad k \ge 1,$$

with

$$p_0(t) = \exp\left\{ -\lambda t \left[1 - \left(\frac{\beta}{\lambda_1+\beta}\right)^{\!\alpha}\,\right]\right\}$$

[(Buchak et al., 2017), (4.1)–(4.2)], (Buchak et al., 2018).

Moment Generating Function

The mgf of $X(t)$ is given by

$$M(u; t) = \mathbb{E}[e^{u X(t)}] = \exp\left\{-\lambda t \left[1 - \beta^\alpha (\beta + \lambda_1(1-e^u))^{-\alpha}\right]\right\}.$$

This expression leverages the subordinator's explicit Laplace exponent and Bochner subordination [(Buchak et al., 2017), (4.12)].

Hitting Times and Ruin Probabilities

Hitting times, or first passage times to state $m$, have the distribution: $$P\{T_m<\infty\} = \frac{\lambda_1^m}{m!(\lambda_1+\beta)^m \left[1-\left(\frac{\beta}{\lambda_1+\beta}\right)^\alpha\right]} \sum_{n=1}^\infty \left(\frac{\beta}{\lambda_1+\beta}\right)^{\alpha n} \frac{\Gamma(\alpha n + m)}{\Gamma(\alpha n)}$$ indicating that the process may never reach state $m$ with strictly positive probability (Buchak et al., 2018).

4. Generalizations and Iterations

Extensions

A linear deterministic drift can be incorporated: $Y(t) = G_{\alpha,\beta}(N_\lambda(t) + a t),$ with Laplace exponent

$$\phi_Y(u) = \alpha a \ln\!\left(1+\frac{u}{\beta}\right) + \lambda\left[1 - (1+u/\beta)^{-\alpha}\right].$$

Time-changing a Poisson process by $Y(t)$ yields distributions analogous to the basic time-changed model, with parameters modified by the drift $a$ (Buchak et al., 2018).

Iterated Compositions

One may construct

$G_1(N(t)), \quad G_2(N(G_1(N(t)))), \ldots$

For $\alpha = 1$, a “semigroup” property holds: iterated compound Poisson–exponential subordinators again yield a process within the same class, with effective parameters composed algebraically. For general $\alpha$, Laplace exponents add hierarchically, leading to stable or quasi-stable limiting behavior in some regimes (Buchak et al., 2018).

5. Inverse Processes and Skellam-Type Constructions

Inverse Subordinators

For $\alpha = 1$, the inverse of a compound Poisson–exponential process, $Y(u) = \inf\{t: S(t)>u\}$, has explicit density and Laplace transform given by Bessel and exponential terms. This structure produces time-changed Poisson processes $N_1(Y(t))$ whose distributions are described by generalized Mittag-Leffler functions [(Buchak et al., 2017), Section 5].

Skellam-Type Differences

Two variants arise:

• Type I (Common Clock): $S_I(t) = N_1(S(t)) - N_2(S(t))$
• Type II (Independent Clocks): $S_{II}(t) = N_1(X_1(t)) - N_2(X_2(t))$, with $X_i(t)$ iid subordinators

Explicit pmfs and mgfs are obtained using integration against the compound Poisson–Gamma laws, with forms involving Wright and Bessel functions [(Buchak et al., 2017), eqs. (4.20), (4.33)].

6. Connections to Anomalous Diffusion and Fractional Calculus

Lower-Incomplete Gamma Subordinator

A related process is defined via the lower-incomplete gamma function: for $a \in (0,1]$,

$$\phi(\eta) = \alpha \gamma(\alpha; \eta), \quad \gamma(\alpha;\eta) = \int_0^\eta w^{\alpha-1}e^{-w} dw$$

[(Beghin et al., 2021), (3.1)]. As $\alpha \to 1$, this recovers the standard Poisson process; for $\alpha < 1$, it serves as a finite-activity approximation to the stable subordinator.

Governing Equations

Time-changing Markov semigroups by the compound Poisson–Gamma or incomplete gamma subordinator yields non-local integro-differential equations, serving as a discrete or mesoscopic analog to fractional-derivative evolution equations

$$\frac{\partial}{\partial t}g(x,t) = \frac{1}{\Gamma(1-\alpha)} \int_\varepsilon^\infty (T_s g(\cdot, t) - g(\cdot, t))(s-\varepsilon)^{-\alpha-1} ds$$

[(Beghin et al., 2021), §6].

Sub-Diffusive Scaling

For a Brownian motion time-changed by the lower-incomplete gamma subordinator, only fractional moments exist; e.g., for $0<p\leq a$,

$$\mathbb{E}[S_a(t)^p] \sim \frac{\Gamma(1 + p/a)}{\Gamma(1+p)} t^{p/a}, \quad t \to \infty$$

and for time-changed fractional Brownian motion,

$$\mathrm{Var}[Z_H(t)] \sim K \, t^{2H/a}, \quad t\to\infty$$

indicating sub-diffusive behavior when $2HBeghin et al., 2021).

7. Applications and Interpretations

Time-changed compound Poisson–Gamma subordinators model heterogeneous arrival or event times in count data, phenomena with clustered or bursty activity, and non-classical diffusive processes. They provide explicit formulas for hitting (ruin) probabilities and enable analytical tractability in iterated and multivariate extensions. Their governing equations link them with fractional and anomalous transport, while their discrete structure supports finite-activity approximations for stable-type jump processes (Buchak et al., 2018, Beghin et al., 2021).

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Summary Table: Core Constructs

Process / Formula	Key Parameters	Defining Equation
Compound Poisson–Gamma subordinator	$\alpha, \beta, \lambda$	$S(t) = \sum_{n=1}^{N_\lambda(t)} G_n$
Lévy measure for subordinator	$\alpha, \beta, \lambda$	$$\nu(du) = \lambda \frac{\beta^\alpha}{\Gamma(\alpha)} u^{\alpha-1} e^{-\beta u} du$$
Time-changed Poisson $N_{\lambda_1}(S(t))$	outer: $\lambda_1$; sub: as above	See section 3, PMF and MGF
Lower-incomplete gamma subordinator	$a, \alpha$	$\psi(\eta) = \alpha \gamma(\alpha;\eta)$

Time-changed compound Poisson–Gamma subordinators thus form a vital bridge between tractable finite-jump processes, classical stochastic calculus, and the domain of fractional, non-local, and anomalous dynamical models (Buchak et al., 2017, Buchak et al., 2018, Beghin et al., 2021).